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1. An analogue of polynomially integrable bodies in even-dimensional spaces

   https://doi.org/10.1016/j.jmaa.2023.127071  

   Agranovsky, M; Koldobsky, A; Ryabogin, D; Yaskin, V. (January 2024, Journal of mathematical analysis and applications)

   A bounded domain $$K \subset R^n$$ is called polynomially integrable ifthe $(n-1)$-dimensional volume of the intersection $$K$$ with a hyperplane $$\Pi$$ polynomially depends on the distance from $$\Pi$$ to the origin. It was proved in \cite{KMY} that there are no such domains with smooth boundary if $$n$$ is even, and if $$n$$ is odd then the only polynomially integrable domains with smooth boundary are ellipsoids. In this article, we modify the notion of polynomial integrability for even $$n$$ and consider bodies for which the sectional volume function is a polynomial up to a factor which is the square root of a quadratic polynomial, or, equivalently, the Hilbert transform of this function is a polynomial. We prove that ellipsoids in even dimensions are the only convex infinitely smooth bodies satisfying this property. 

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2. Near-Polynomially Competitive Active Logistic Regression

   Zhou, Y; Price, E; Nguyen, T (March 2025, https://doi.org/10.48550/arXiv.2503.05981)

   We address the problem of active logistic regression in the realizable setting. It is well known that active learning can require exponentially fewer label queries compared to passive learning, in some cases using $$\log \frac{1}{\eps}$$ rather than $$\poly(1/\eps)$$ labels to get error $$\eps$$ larger than the optimum. We present the first algorithm that is polynomially competitive with the optimal algorithm on every input instance, up to factors polylogarithmic in the error and domain size. In particular, if any algorithm achieves label complexity polylogarithmic in $$\eps$$, so does ours. Our algorithm is based on efficient sampling and can be extended to learn more general class of functions. We further support our theoretical results with experiments demonstrating performance gains for logistic regression compared to existing active learning algorithms. 

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3. On polynomially integrable convex bodies

   Koldobsky, Alexander; Merkurjev, Alexander; Yaskin, Vladyslav (October 2017, Advances in mathematics)
4. Convex Union Representability and Convex Codes

   https://doi.org/10.1093/imrn/rnz055  

   Jeffs, R Amzi; Novik, Isabella (April 2019, International Mathematics Research Notices)

   null (Ed.)

   Abstract We introduce and investigate $$d$$-convex union representable complexes: the simplicial complexes that arise as the nerve of a finite collection of convex open sets in $${\mathbb{R}}^d$$ whose union is also convex. Chen, Frick, and Shiu recently proved that such complexes are collapsible and asked if all collapsible complexes are convex union representable. We disprove this by showing that there exist shellable and collapsible complexes that are not convex union representable; there also exist non-evasive complexes that are not convex union representable. In the process we establish several necessary conditions for a complex to be convex union representable such as that such a complex $$\Delta $$ collapses onto the star of any face of $$\Delta $$, that the Alexander dual of $$\Delta $$ must also be collapsible, and that if $$k$$ facets of $$\Delta $$ contain all free faces of $$\Delta $$, then $$\Delta $$ is $(k-1)$-representable. We also discuss some sufficient conditions for a complex to be convex union representable. The notion of convex union representability is intimately related to the study of convex neural codes. In particular, our results provide new families of examples of non-convex neural codes. 

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5. Coarse Distance from Dynamically Convex to Convex

   https://doi.org/10.1093/imrn/rnaf179  

   Dardennes, Julien; Gutt, Jean; Ramos, Vinicius_G_B; Zhang, Jun (June 2025, International Mathematics Research Notices)

   Abstract Chaidez and Edtmair have recently found the first examples of dynamically convex domains in $$\mathbb{R}^{4}$$ that are not symplectomorphic to convex domains, answering a long-standing open question. In this paper, we discover new examples of such domains without referring to Chaidez–Edtmair’s methods. We show a stronger result: that these domains are arbitrarily far from the set of convex domains in $$\mathbb{R}^{4}$$ with respect to the coarse symplectic Banach–Mazur distance. 

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